1. Unpredictable (closest to mathematical
definition); weird. "The system's been behaving pretty
randomly." 2. Assorted; undistinguished. "Who was at the
conference?" "Just a bunch of random business types."
3. (pejorative) Frivolous; unproductive; undirected. "He's just a
random loser." 4. Incoherent or inelegant; poorly chosen; not
well organized. "The program has a random set of misfeatures."
"That's a random name for that function." "Well, all the names
were chosen pretty randomly." 5. In no particular order, though
deterministic. "The I/O channels are in a pool, and when a file
is opened one is chosen randomly." 6. Arbitrary. "It generates
a random name for the scratch file." 7. Gratuitously wrong, i.e.,
poorly done and for no good apparent reason. For example, a
program that handles file name defaulting in a particularly useless
way, or an assembler routine that could easily have been coded
using only three registers, but redundantly uses seven for values
with non-overlapping lifetimes, so that no one else can invoke it
without first saving four extra registers. What randomness!
8. n. A random hacker; used particularly of high-school
students who soak up computer time and generally get in the way.
9. n. Anyone who is not a hacker (or, sometimes, anyone not known
to the hacker speaking); the noun form of sense 2. "I went to the
talk, but the audience was full of randoms asking bogus
questions". 10. n. (occasional MIT usage) One who lives at
Random Hall. See also J. Random, some random X.
11. [UK] Conversationally, a non sequitur or something similarly
out-of-the-blue. As in: "Stop being so random!" This sense
equates to `hatstand', taken from the Viz comic character "Roger
Irrelevant - He's completely Hatstand."

Algorithmic Information Theory defines the extent to which a sequence of numbers is random by the length of the shortest algorithm (i.e. programme) that outputs it. So "11111111111111" could be output by a little programme like
for i=1 to 15
{print "1"}

while the shortest programme to output "346357538323523627567" might actually be
print "346357538323523627567"

so the second sequence is "more random" than the first. In fact it defines a "random sequence" as one for which the shortest algorithm is just "print the sequence".
You can make this idea more concrete by considering a Turing machine made to output the sequence rather than just a randomprogramming language.
Note that this ties in nicely with our common understanding of randomness - if there's a nice pattern there, then you can exploit that to compress the sequence into a little algorithm, and hence it is not by this definition random.

"Random" was a word that was popular to use for about a decade or so, in the late 1990s and early 2000s. Beyond being an oft-used and abused word, it was an entire aesthetic, where incongruous things were matched up with each other. Often these "random" things were not as "random" as they seemed to be, being drawn from a rather limited lexicon of wackyness. See, for example, Mr. T ate my balls. The aesthetic of "Random" was a major force in the early development of the world wide web, with many examples from this site'searlyyears.

I believe that "random" as a slang term has probably receded, although given the psychology and sociology of late adolescence, the underlying idea probably has not. But perhaps randomness can not be seen as that great of a thing where so many ukulele covers of 80s pop songs exist on youtube, and where no less of an arbiter of culture as xkcd has declared it obsolete.

For courageously the two kings newly fought with great random and force.
E. Hall.

2.

A roving motion; course without definite direction; want of direction, rule, or method; hazard; chance; -- commonly used in the phrase at random, that is, without a settled point of direction; at hazard.

Counsels, when they fly
At random, sometimes hit most happily.
Herrick.

Going at random or by chance; done or made at hazard, or without settled direction, aim, or purpose; hazarded without previous calculation; left to chance; haphazard; as, a random guess.

Some random truths he can impart.
Wordsworth.

So sharp a spur to the lazy, and so strong a bridle to the random.
H. Spencer.

Random courses Masonry, courses of unequal thickness. -- Random shot, a shot not directed or aimed toward any particular object, or a shot with the muzzle of the gun much elevated. -- Random work Masonry, stonework consisting of stones of unequal sizes fitted together, but not in courses nor always with flat beds.